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＜電子ブック＞
A Stability Technique for Evolution Partial Differential Equations : A Dynamical Systems Approach / by Victor A. Galaktionov, Juan Luis Vázquez
(Progress in Nonlinear Differential Equations and Their Applications. ISSN:23740280 ; 56)

版	1st ed. 2004.
出版者	(Boston, MA : Birkhäuser Boston : Imprint: Birkhäuser)
出版年	2004
本文言語	英語
大きさ	XIX, 377 p : online resource
著者標目	*Galaktionov, Victor A author Vázquez, Juan Luis author SpringerLink (Online service)
件　名	LCSH:Differential equations LCSH:Mathematical analysis LCSH:Mechanics, Applied LCSH:Solids LCSH:Fluid mechanics FREE:Differential Equations FREE:Analysis FREE:Solid Mechanics FREE:Engineering Fluid Dynamics
一般注記	Introduction: A Stability Approach and Nonlinear Models -- Stability Theorem: A Dynamical Systems Approach -- Nonlinear Heat Equations: Basic Models and Mathematical Techniques -- Equation of Superslow Diffusion -- Quasilinear Heat Equations with Absorption. The Critical Exponent -- Porous Medium Equation with Critical Strong Absorption -- The Fast Diffusion Equation with Critical Exponent -- The Porous Medium Equation in an Exterior Domain -- Blow-up Free-Boundary Patterns for the Navier-Stokes Equations -- The Equation ut = uxx + uln2u: Regional Blow-up -- Blow-up in Quasilinear Heat Equations Described by Hamilton-Jacobi Equations -- A Fully Nonlinear Equation from Detonation Theory -- Further Applications to Second- and Higher-Order Equations -- References -- Index common feature is that these evolution problems can be formulated as asymptoti­ cally small perturbations of certain dynamical systems with better-known behaviour. Now, it usually happens that the perturbation is small in a very weak sense, hence the difficulty (or impossibility) of applying more classical techniques. Though the method originated with the analysis of critical behaviour for evolu­ tion PDEs, in its abstract formulation it deals with a nonautonomous abstract differ­ ential equation (NDE) (1) Ut = A(u) + C(u, t), t > 0, where u has values in a Banach space, like an LP space, A is an autonomous (time-independent) operator and C is an asymptotically small perturbation, so that C(u(t), t) ~ ° as t ~ 00 along orbits {u(t)} of the evolution in a sense to be made precise, which in practice can be quite weak. We work in a situation in which the autonomous (limit) differential equation (ADE) Ut = A(u) (2) has a well-known asymptotic behaviour, and we want to prove that for large times the orbits of the original evolution problem converge to a certain class of limits of the autonomous equation. More precisely, we want to prove that the orbits of (NDE) are attracted by a certain limit set [2* of (ADE), which may consist of equilibria of the autonomous equation, or it can be a more complicated object HTTP:URL=https://doi.org/10.1007/978-1-4612-2050-3

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